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Register a new userCausal simplicity and (maximal) null pseudoconvexity
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We consider pseudoconvexity properties in Lorentzian and Riemannian manifolds and their relationship in static spacetimes. We provide an example of a causally continuous and maximal null pseudoconvex spacetime that fails to be causally simple. Its Riemannian factor provides an analogous example of a manifold that is minimally pseudoconvex, but fails to be convex.
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The Averaged Null Energy Condition (ANEC) states that the integral along a complete null geodesic of the projection of the stress-energy tensor onto the tangent vector to the geodesic cannot be negative. ANEC can be used to rule out spacetimes with exotic phenomena, such as closed timelike curves, superluminal travel and wormholes. We prove that ANEC is obeyed by a minimally-coupled, free quantum scalar field on any achronal null geodesic (not two points can be connected with a timelike curve) surrounded by a tubular neighborhood whose curvature is produced by a classical source. To prove ANEC we use a null-projected quantum inequality, which provides constraints on how negative the weighted average of the renormalized stress-energy tensor of a quantum field can be. Starting with a general result of Fewster and Smith, we first derive a timelike projected quantum inequality for a minimally-coupled scalar field on flat spacetime with a background potential. Using that result we proceed to find the bound of a quantum inequality on a geodesic in a spacetime with small curvature, working to first order in the Ricci tensor and its derivatives. The last step is to derive a bound for the null-projected quantum inequality on a general timelike path. Finally we use that result to prove achronal ANEC in spacetimes with small curvature.
Reasonable spacetimes are non-compact and of dimension larger than two. We show that these spacetimes are globally hyperbolic if and only if the causal diamonds are compact. That is, there is no need to impose the causality condition, as it can be deduced. We also improve the definition of global hyperbolicity for the non-regular theory (non $C^{1,1}$ metric) and for general cone structures by proving the following convenient characterization for upper semi-continuous cone distributions: causality and the causally convex hull of compact sets is compact. In this case the causality condition cannot be dropped, independently of the spacetime dimension. Similar results are obtained for causal simplicity.
In this work we give a complete picture of how to in a direct simple way define the mass at null infinity in harmonic coordinates in three different ways that we show satisfy the Bondi mass loss law. The first and second way involve only the limit of metric (Trautman mass) respectively the null second fundamental forms along asymptotically characteristic surfaces (asymptotic Hawking mass) that only depend on the ADM mass. The last in an original way involves construction of special characteristic coordinates at null infinity (Bondi mass). The results here rely on asymptotics of the metric derived in [24].
The possibility of finding the measurable maximal energy and the minimal time interval is discussed in different quantum aspects. It is found that the linear generalized uncertainty principle (GUP) approach gives a non-physical result. Based on large scale Schwarzshild solution, the quadratic GUP approach is utilized. The calculations are performed at the shortest distance, at which the general relativity is assumed to be a good approximation for the quantum gravity and at larger distances, as well. It is found that both maximal energy and minimal time have the order of the Planck time. Then, the uncertainties in both quantities are accordingly bounded. Some physical insights are addressed. Also, the implications on the physics of early Universe and on quantized mass are outlined. The results are related to the existence of finite cosmological constant and minimum mass (mass quanta).
We define resonances for finitely perturbed quantum walks as poles of the scattering matrix in the lower half plane. We show a resonance expansion which describes the time evolution in terms of resonances and corresponding Jordan chains. In particular, the decay rate of the survival probability is given by the imaginary part of resonances and the multiplicity. We prove generic simplicity of the resonances, although there are quantum walks with multiple resonances.
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